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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X20,10,21,9 X16,8,17,7 X18,14,19,13 X14,20,15,19 X22,16,11,15 X10,18,5,17 X8,22,9,21 X2536 X4,11,1,12

Gauss Code:

{{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {11, -2, 5, -6, 7, -4, 8, -5, 6, -3, 9, -7}}

Jones Polynomial:

q-3 - 3q-2 + 9q-1 - 14 + 21q - 23q2 + 25q3 - 21q4 + 16q5 - 10q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 - q-6 + 5q-4 + 3 + 9q2 + q4 + 9q6 + 2q8 + 3q10 + 3q12 - 5q14 + 3q16 - 4q18 - 3q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z-2 - 2a-6 - a-6z2 - a-6z4 + 4a-4z-2 + 9a-4 + 6a-4z2 + 2a-4z4 + a-4z6 - 5a-2z-2 - 11a-2 - 8a-2z2 - a-2z4 + a-2z6 + 2z-2 + 3 - z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 - 4a-8z4 + 4a-8z6 + a-7z-1 - 4a-7z + 9a-7z3 - 14a-7z5 + 9a-7z7 - a-6z-2 + 3a-6 - 9a-6z2 + 19a-6z4 - 22a-6z6 + 12a-6z8 + 5a-5z-1 - 19a-5z + 29a-5z3 - 20a-5z5 - 2a-5z7 + 8a-5z9 - 4a-4z-2 + 15a-4 - 29a-4z2 + 52a-4z4 - 54a-4z6 + 20a-4z8 + 2a-4z10 + 9a-3z-1 - 33a-3z + 45a-3z3 - 21a-3z5 - 14a-3z7 + 13a-3z9 - 5a-2z-2 + 20a-2 - 38a-2z2 + 48a-2z4 - 43a-2z6 + 14a-2z8 + 2a-2z10 + 5a-1z-1 - 18a-1z + 29a-1z3 - 22a-1z5 + 5a-1z9 - 2z-2 + 8 - 15z2 + 16z4 - 14z6 + 6z8 + 3az3 - 6az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   7 1   j = 11                 9 3     j = 9               12 7       j = 7             13 9         j = 5           12 14           j = 3         9 11             j = 1       6 13               j = -1     3 8                 j = -3     6                   j = -5 1 3                     j = -7 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 423]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 423]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 10, 21, 9], X[16, 8, 17, 7], > X[18, 14, 19, 13], X[14, 20, 15, 19], X[22, 16, 11, 15], X[10, 18, 5, 17], > X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, > {11, -2, 5, -6, 7, -4, 8, -5, 6, -3, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 9 2 3 4 5 6 7 8 -14 + q - -- + - + 21 q - 23 q + 25 q - 21 q + 16 q - 10 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 5 2 4 6 8 10 12 14 16 3 + q - q + -- + 9 q + q + 9 q + 2 q + 3 q + 3 q - 5 q + 3 q - 4 q 18 20 22 24 > 4 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 423]][a, z]
Out[8]=	2 2 2 2 9 11 2 2 1 4 5 2 z 6 z 8 z 3 - -- + -- - -- + a + -- - ----- + ----- - ----- - z - -- + ---- - ---- + 6 4 2 2 6 2 4 2 2 2 6 4 2 a a a z a z a z a z a a a 4 4 4 6 6 2 2 4 z 2 z z z z > a z - 2 z - -- + ---- - -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 423]][a, z]
Out[9]=	3 15 20 2 2 1 4 5 1 5 9 5 8 + -- + -- + -- - a - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- - 6 4 2 2 6 2 4 2 2 2 7 5 3 a z a a a z a z a z a z a z a z a z 2 2 2 3 4 z 19 z 33 z 18 z 2 9 z 29 z 38 z 2 2 z > --- - ---- - ---- - ---- - 15 z - ---- - ----- - ----- + 3 a z - -- + 7 5 3 a 6 4 2 9 a a a a a a a 3 3 3 3 4 4 4 9 z 29 z 45 z 29 z 3 4 4 z 19 z 52 z > ---- + ----- + ----- + ----- + 3 a z + 16 z - ---- + ----- + ----- + 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 5 48 z 2 4 z 14 z 20 z 21 z 22 z 5 6 > ----- - 3 a z + -- - ----- - ----- - ----- - ----- - 6 a z - 14 z + 2 9 7 5 3 a a a a a a 6 6 6 6 7 7 7 4 z 22 z 54 z 43 z 2 6 9 z 2 z 14 z 7 > ---- - ----- - ----- - ----- + a z + ---- - ---- - ----- + 3 a z + 8 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 12 z 20 z 14 z 8 z 13 z 5 z 2 z 2 z > 6 z + ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 3 6 3 8 6 q 3 13 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 11 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 12 q t + 14 q t + 13 q t + 9 q t + 12 q t + 7 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a423

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